What type of wheels do we use for precise movement and turning? We are considering a tri-wheel design or a tracked design.
for starters, you need to decide on your robot approximate specifications, like the weight,dimensions, and the position of wheels etc. A tri-wheel design would work fine for you (i presume NERC is the competition), considering its very simple to implement.http://www.societyofrobots.com/mechanics_chassisconstruction.shtml
Check out the above link, and scroll down to the "wheel basics" section. That should give you some basic idea on some parameters to keep in mind while selecting wheels. Since the robot needs to navigate a wall maze, you need to make tight 90 and 180 deg turns, just make sure the wheels don't skid on the floor mat. although i find them unnecessary in this case, you may try adding encoders for extra precision.